Abstract of the extended version

Henselian Function Fields and Tame Fields

of my thesis

Franz-Viktor Kuhlmann

Results on the algebra of valued fields.

In the theory of algebraic function fields (which we will simply call function fields), the valued function fields (where the valuation is not necessarily trivial on the base field) play an important role. For instance, constant reduction of function fields is given by valuations with certain properties that we will discuss later. One can read off important information about a valued function field from its henselizations. Hence, henselizations of valued function fields appear in a natural way when valued function fields are studied. This is important, if for instance the valuation is nontrivial on the base field or the transcendence degree is greater than 1 since then the completion is in general not an adequate device for the study of such valued function fields. For example, the question arises whether a given valued function field F has the property that every finite extension E of F is defectless, i.e. satisfies the fundamental equality

n = ∑ e_{i}f_{i}  (1)

where n = [E : F] is the degree of the extension and e_{i}, f_{i} are ramification index and inertia degree of the finitely many extensions of the valuation of F to the extension field E. If a valued field has this property, then we will call it a defectless field. Note that every valued field of residue characteristic 0 is a defectless field; this follows from the Lemma of Ostrowski. Now if F^{h} is a henselization of F, then it is known that F is a defectless field if and only F^{h} is a defectless field. Though F^{h} has the disadvantage that it is not itself a function field (if the given valuation is nontrivial), it has the advantage that it satisfies Hensel’s Lemma and that the valuation admits a unique extension to every algebraic extension of F^{h}. These facts can be very helpful for the study of the given valuation. Note that a henselization is an immediate extension; an extension L|K of valued fields is called immediate, if v(K) = v(L) and \overline{K} = \overline{L}. Here v(K), v(L) denote the value groups and \overline{K}, \overline{L} denote the residue fields of K and L; we will also write K/v instead of \overline{K} if several valuations are considered. Henselian defectless fields are the same as algebraically complete fields, i.e. fields that have a unique extension of the valuation to every finite extension field which moreover satisfies the fundamental equality

n = e · f  (2)

(this is just the version of (1) for the case of a unique extension).

Henselizations of valued function fields will be called “henselian function fields” though they are in general not function fields, as we have mentioned already. Both, henselian function fields and defectless fields play the central role in our investigations. Without any additional assumptions there is no hope for a simple structure theory of henselian function fields. But under certain assumptions which are common in the theory of valued function fields, we prove that henselian function fields have a rather simple structure.

Let us discuss two special cases to illustrate the significance of defectless fields and the possible assumptions on the henselian function field F |K. The valuation on both F and K will be denoted by v. The first special case is given by the hypothesis that F is the henselization of a function field of transcendence degree 1 and that F |K is an immediate extension. Let us take an arbitrary element x ∈ F which is transcendental over K. Choosing a henselization K(x)^{h} of K(x) inside the henselian field F, we find that F |K(x)^{h} is a finite extension which is immediate since F |K is immediate. Note that the henselian field K(x)^{h} admits a unique extension of the valuation v to the algebraic extension field F. If in this situation we would know that K(x)^{h} is a defectless field (e.g. if char(\overline{K(x)^{h}}) = 0), then the finite immediate extension F |K(x)^{h} would necessarily be trivial:

[F : K(x)^{h}] = e · f = (v(F) : v(K(x)^{h}) · [\overline{F} : \overline{K(x)^{h}}] = 1 · 1 = 1 ;

here we have used equation (2). Consequently,

F = K(x)^{h} ,

the henselization of a rational function field; such a henselian function field will be called henselian rational function field. The assumption “char(\overline{K(x)}) = 0” can also be expressed as an assumption on the base field K; indeed, it holds if

– char(\overline{K}) = 0. There are other possible sufficient assumptions on K:

– (K, v) is a (formally) ℘–adic field, or, more generally,

– (K, v) is a finitely ramified field, i.e. v(K) admits a least positive element and there is a prime p such that v(p · 1_{K}) is a multiple of this element; in this case, char(K) = 0 and char(\overline{K}) = p. The structure of these fields is slightly more general than that of formally ℘–adic fields. The difference is that the residue fields may be arbitrary not necessarily finite fields of characteristic p > 0. It is known that all finitely ramified fields are defectless. On the other hand, it can be read off from the definition that an immediate extension of a finitely ramified field is again finitely ramified. So if we assume that the base field K is finitely ramified, the same will hold for the immediate extension K(x)^{h} and again, we obtain that K(x)^{h} is defectless and that F = K(x)^{h}. Note that the henselian field F contains a henselization K^{h} of K, so we can assume from the start that the base field K is henselian.

Other possible assumptions on the base field may fail in this situation, even if they are as common as the following one:

– K is algebraically closed. Note that every algebraically closed valued field is trivially a defectless field. Another possible condition is known from the paper “Maximal fields with valuations” [KAP] of Kaplansky:

– (K, v) is a henselian defectless Kaplansky–field; a Kaplansky–field is a valued field satisfying Kaplansky’s hypothesis (A). Hypothesis (A) was later shown by Whaples [WHA] and, independently, by Delon [DEL] to mean that (Kf 1) the value group v(K) is p–divisible (p = char(\overline{K}) > 0),

(Kf 2) the residue field \overline{K} is perfect (i.e. it allows no inseparable algebraic extensions),

(Kf 3) every finite separable extension of the residue field has degree not divisible by p.

In general, henselian Kaplansky–fields are not defectless fields. Even if the base field K is a henselian defectless Kaplansky–field, an immediate henselian function field may not be defectless, though it is also a Kaplansky–field (because the conditions for Kaplansky– fields only talk about the value group and the residue field). Similarly, there exist immediate henselian function fields over algebraically closed base fields of positive residue characteristic which are not defectless fields. In both cases it turns out that it is not a sufficient approach if we choose the transcendental element x without any care. We show that it is indeed possible to choose x well; we do this e.g. under the assumption that K is a perfect henselian defectless field of positive characteristic. This class of fields includes the algebraically closed valued fields of positive characteristic and the henselian defectless Kaplansky–fields of positive characteristic. Recall that “henselian defectless” is the same as “algebraically complete”. For perfect fields of positive characteristic and for Kaplansky–fields of positive characteristic, “algebraically complete” is equivalent to algebraically maximal which means that the valued field has no immediate algebraic extensions. Our result for algebraically maximal perfect fields of positive characteristic is the following:

Theorem 1 Let K be an algebraically maximal perfect field of positive characteristic and F an immediate henselian function field over K. If its transcendence degree over K is 1, then F is a henselian rational function field over K. In the general case of transcendence degree ≥ 1, there is a finite immediate extension F′ of F such that F′ is a henselian rational function field over K.

We show that this theorem does not remain true if the condition “perfect” is dropped without a substitute. Note that if we replace the condition on the characteristic by “char(\overline{K}) = 0” then the theorem remains true: for every transcendence basis T of F |K we have F = K(T )^{h}. This is proved as described above in the case of transcendence degree 1. In contrast to this short proof, the proof of Theorem 1 requires much technical effort. We give normal forms for certain classes of immediate extensions.

An extension of a henselian field F is called tame extension if it lies in the absolute ramification field F^{r} of F. This field F^{r} is the ramification field of the normal separable extension F^{sep}|F, where F^{sep} is the separable-algebraic closure of F. Further, F is called tame if every algebraic extension is tame, i.e., its absolute ramification field F^{r} is algebraically closed. Immediate algebraic extensions of F are purely wild, that is, linearly disjoint from the maximal tame extension F^{r}|F (cf. [KPR]). Therefore, we start our search for normal forms with a description of the structure of minimal purely wild algebraic extensions of henselian fields of positive characteristic, given by F. Pop. The theory of purely wild algebraic extensions was developed by M. Pank [PAN1], [PAN2] and is summarized in [KPR]. We use it at several points as a basic tool for our investigations.

In particular, it is used to characterize the algebraically complete perfect fields of positive characteristic to be just the tame fields of positive characteristic.

The class of tame fields comprises almost all assumptions on F that we have discussed so far:

– algebraically closed valued fields

– henselian fields of residue characteristic 0

– algebraically maximal Kaplansky fields

– algebraically maximal perfect fields of positive characteristic are tame fields. The henselian formally ℘–adic fields and the henselian finitely ramified fields are missing in this list. But their valuation admits a finest coarsening under which they become a tame field having a residue field of characteristic 0 with a discrete henselian valuation of residue characteristic p > 0. This special structure guarantees that they are defectless fields.

We prove that the above assertion of Theorem 1 holds for all tame fields:

Theorem 2 Let K be a tame field and F an immediate function field over K. If its transcendence degree over K is 1, then F^{h} is a henselian rational function field over K^{h}. In the general case of transcendence degree ≥ 1, given any immediate extension N of F which is a tame field (such an extension does always exist), there is a finite immediate extension F_{1} of F within N such that F_{1}^{h} is a henselian rational function field over K.

For this generalization, we have to treat the case of a tame base field (K, v) of “mixed characteristic”, i.e. char(K) = 0 and char(\overline{K}) = p > 0. Since in the mixed characteristic case, we do not know an analogue of the lemma of Pop, we start the proof assuming K to be algebraically closed, and then we involve a “pull down principle for henselian rationality through tame extensions”. This in turn uses a valuation theoretical property that characterizes the Galois groups of tame Galois extensions. In both characteristic cases, our proof requires an important property of tame fields:

Lemma 3 Let L be a tame field and K ⊂ L a relatively algebraically closed subfield. If in addition \overline{L}|\overline{K} is an algebraic extension, then K is also a tame field and moreover, v(K) is pure in v(L) and \overline{K} = \overline{L}.

This lemma is used several times in the proof of Theorem 2 and in the applications of this theorem. It allows us to break up a given transcendental extension of tame fields into an immediate and an “anti–immediate” extension of tame fields (the meaning of the latter notion is explained below). Moreover, it shows the immediate extension to be a tower of extensions of transcendence degree 1 of tame fields. Note that the lemma does not remain true if one replaces “tame” by “algebraically complete” or drops the condition on the residue fields.

The above results are generalized to separably tame fields; these are henselian fields for which every separable–algebraic extension is tame. In characteristic 0, they coincide with the tame fields; but in positive characteristic, they form a bigger (elementary) class. They are connected to tame fields as follows: (K, v) is a separably tame field if and only if its perfect closure (\sqrt{K}, v) is a tame field. In this case, (\sqrt{K}, v) lies in the completion of (K, v).

The second special case of henselian function fields is somewhat the complement of the first: it is the “anti–immediate” case. We have to make this precise. The assumption is that the henselian function field F |K has no transcendence defect, i.e.

trdeg(F |K) = trdeg(\overline{F} |\overline{K}) + rr(v(F)/v(K))

where rr(v(F)/v(K)) denotes the rational rank (i.e. the maximal number of rationally independent elements) of the abelian group v(F)/v(K). The assumption is fulfilled if constant reduction of function fields is studied: in this case we have by definition that

trdeg(F |K) = trdeg(\overline{F} |\overline{K})

which implies rr(v(F)/v(K)) = 0. For tame fields, extensions without transcendence defect are indeed the complement to immediate extensions, in the following sense: with the help of Lemma 3 it can be shown that every extension L|K of tame fields admits a subextension L′|K such that L′ is a tame field, L′|K is an extension without transcendence defect, and L|L′ is an immediate extension. In this situation, every henselization of a finitely generated subextension L_{0} of L′|K is a henselian function field without transcendence defect; we use this fact when studying the model theory of tame fields, as we will describe below.

For henselian function fields without transcendence defect, we prove:

Theorem 4 Let F |K be a henselian function field without transcendence defect. If K is a defectless field then F is a defectless field.

With the help of this theorem it is easy to show that a henselian function field F |K without transcendence defect over a defectless base field K is not only itself a defectless field, but also that it has a simple structure: it is a finite defectless extension of a henselian rational function field. Depending on the value group and the residue field of F, the structure may even be much simpler. If for instance, \overline{F} = \overline{K} or \overline{F} |\overline{K} is a rational function field and if v(K) is pure in v(F), then F is a henselian rational function field. If \overline{F} |\overline{K} is separable and v(K) pure in v(F) then F^{h} is “almost” a henselian rational function field: it is a finite inert extension of a henselian rational function field. These results together with Theorem 2 generalize a theorem that was proved by M. Matignon in an unpublished paper [MAT1] and announced, together with a sketch of the proof, in [MAT1a], cf. Théorème 3, p. 7, and in [MAT1b]. (Matignon proves that the completion of F is the completion of a suitable rational function field, under the following additional assumptions: v(F) is of rank 1, i.e. archimedean, \overline{F} = \overline{K}, and K is algebraically closed and complete with respect to the valuation v.)

Theorem 4 is a generalization of a theorem of Grauert and Remmert [GR] which works with completions instead of henselizations and is restricted to the case of algebraically closed complete base fields of rank 1 (the value group is archimedean). A generalization of the Grauert–Remmert Theorem was given by Gruson [GRU], an improved presentation of it can be found in the book [BGR] of Bosch, Güntzer and Remmert. The proof uses methods of nonarchimedean analysis. Further generalizations are due to M. Matignon and J. Ohm; see also [GMP]. Ohm [OHM3] arrived independently of our thesis at a version of Theorem 4 which is restricted to the case trdeg(F |K) = trdeg(\overline{F} |\overline{K}). The idea of Ohm’s proof is to reduce the general case of valuations of arbitrary rank to the case of rank 1 and to give an auxiliary argument for the transition from completions to henselizations. In this way, he is able to deduce his theorem from [BGR], Proposition 3, p. 215 (more precisely, from the generalized version of this proposition which is proved but not stated in [BGR]).

In contrast to this approach, we give a new proof which replaces the analytic methods of [BGR] by valuation theoretical arguments. Such arguments appear to be more adequate for a theorem that is of valuation theoretical nature. Nevertheless, for the case of trdeg(\overline{F} |\overline{K}) > 0 there is an argument in our proof which may be rather unexpected. We have to use the existence of certain bases for the function field \overline{F} |\overline{K} which we will call Frobenius–closed bases. These are bases where every p–th power of a basis element is also contained in the basis (here p denotes the characteristic of \overline{K}). We show the existence of such bases. It seems that the use of such bases can replace arguments of algebraic geometry that were used in the “classical” approaches; it would certainly be an interesting task to determine the connection between both methods.

Our proof of Theorem 4 also uses reduction to valuations of rank 1; further reduction leads to Galois extensions of degree p = char(\overline{K}) > 0 over a henselian “almost” rational function field F_{0} over an algebraically closed base field. Such a Galois extension is an Artin–Schreier–extension generated by an element ϑ with ϑ^{p} − ϑ = a ∈ F_{0} if char(F) = p, and a Kummer–extension (by a p–th root of an element a ∈ F_{0}) if char(F) = 0. We deduce a normal form for a which allows us to read off that the extension is defectless. A normal form for a can also be read as a result on the structure of the group F_{0}/℘(F_{0}) resp. F_{0}^{×}/(F_{0}^{×})^{p}. Such results can already be found in the work of Hasse, Whaples and others, in the paper “Eliminating wild ramification” of H. Epp [EPP], and in Matignon’s proof of Theorem 4.

A main application of generalizations of the Grauert–Remmert Theorem is that they enable us to define a defect for valued function fields, which has good properties if the function field has no transcendence defect. For an arbitrary algebraic extension L|K of valued fields, the defect is defined to be

d(L|K) = d(L^{h}|K^{h}) = \frac{[L^{h} : K^{h}]}{(v(L) : v(K)) · [\overline{L} : \overline{K}]}

if the extension L^{h}|K^{h} is finite; here the henselization K^{h} is always chosen inside of L^{h}. The defect is a measure for the deviation from the property of being defectless. It is also possible to use the respective ramification fields in the place of the henselizations in order that the definition works in all cases where the extension L^{r}|K^{r} of the ramification fields is finite; if in addition also L^{h}|K^{h} is finite then both definitions coincide. Now the defect of a valued function field F |K can be defined to be

d(F |K) := {sup \atop T} d(F |K(T )) ,

where the supremum runs over all transcendence bases T of F |K. For the case that F |K has no transcendence defect, we show that this supremum is a finite number and moreover, that it is equal to d(F |K(T )) for every valuation transcendence basis T of F |K. A valuation transcendence basis is a transcendence basis with special valuation theoretical properties, comparable to the additional properties of a basis to be a valuation basis. This result was independently obtained by Ohm [OHM3] in the case where trdeg(F |K) = trdeg(\overline{F} |\overline{K}). He calls it “Independence Theorem” since it shows the independence of the defect from the chosen valuation transcendence basis.

We also carry through a closer investigation of the defect of function fields without transcendence defect, in particular, we connect it to certain algebraic extensions of the base field K and their defect.

Matignon [MAT2] and later Green, Matignon and Pop [GMP] use a different definition of the defect of valued function fields for the formulation and the proof of important genus reduction inequalities. These inequalities connect the genus of a function field F |K of transcendence degree 1 (where K is the exact constant field) with the genuses of the reduced function fields

F/v_{i} | K/v (1 ≤ i ≤ s)

where the valuations v_{i} are distinct extensions of a given valuation v on K, all valuations being of rank 1 and such that all extensions F/v_{i} | K/v are transcendental, hence finitely generated of transcendence degree 1. The defect used here coincides with the “henselian” defect that we have introduced above, if (K, v) is henselian (see [GMP] for details). Generalizations of Matignon’s results to valuations of arbitrary rank (cf. [GMP]) have led to an increased interest in several possible notions of the defect for valuations of arbitrary rank (and the corresponding independence results).

Matignon and Ohm (cf. [MAT2],[OHM3]) have used a completion defect which measures the defect (for valuations of rank 1) using completions instead of henselizations; it can be analogously defined for valued function fields. This defect is always less or equal to the henselian defect. In [MAT2], Matignon has shown an independence result for the completion defect of valued function fields (for valuations of rank 1). Apart from the defect for general function fields without transcendence defect that we discussed already, we study a possible generalization of the completion defect. Since for valuations of arbitrary rank, the completion is in general not henselian, we measure the defect over the completion of the henselization: we define the completion defect to be

dm (L|K) := d(L^{hc}|K^{hc}) ,

where “.^{hc}” denotes the completion of a henselization. This defect coincides with Matignon’s and Ohm’s completion defect if the valuation is of rank 1.

We characterize those extensions for which the completion defect is equal to the henselian defect, and compute the defect quotient which are be defined to be the quotient of the henselian defect and the completion defect:

dc (L|K) := \frac{d(L|K)}{dm (L|K)} = \frac{[L : K]_{insep}}{[L^{c} : K^{c}]_{insep}}

where “.^{c}” denotes the completion. For the proof, a special characterization of elements of henselizations is required.

The completion defect of valued function fields without transcendence defect can be defined similarly as it was done for the henselian defect. We prove the Independence Theorem for the completion defect and the defect quotient. Furthermore, we study the behaviour of henselian defect, completion defect and defect quotient under a given decomposition of the valuation.

A valued field K is called c–defectless (resp. q–defectless) if every finite extension L|K has trivial completion defect (resp. defect quotient). As an analogue to Theorem 4, we prove for the defect quotient: Let F be a henselian function field without transcendence defect over K. Assume that K is a q–defectless field or that v(K) is not cofinal in v(F). In both cases, F is a q–defectless field.

The properties of being “c–defectless” or “q–defectles” are weaker than “defectless”. Another pair of weaker properties are inseparably defectless (= every finite purely inseparable extension is defectless) and separably defectless (= every finite separable extension is defectless). For the first property, a theorem which corresponds to Theorem 4 is the following: Let F |K be a henselian function field without transcendence defect. If K is inseparably defectless then F is inseparably defectless.

The corresponding theorem for the second property is:

Theorem 5 Let F |K be a henselian function field without transcendence defect. If K is separably defectless and v(K) is cofinal in v(F) then F is separably defectless.

For the proof of this theorem, we first study the properties of separably defectless fields. This enables us to characterize these fields as just being the c–defectless fields, and the theorem is proved on the basis of this identification. Since the notion “separably defectless” appears to be quite natural, this identification also shows that it is a reasonable approach to measure the defect over the completion of the henselization. This in turn means that completions are of interest even for valuations of arbitrary rank; indeed, they eliminate immediate extensions of a very special form (which in some sense are not as “wild” as the other immediate algebraic extensions of henselian fields).

We characterize henselian defectless fields. The result which is used for the proof of Theorem 5 is the following: Let K be a separable–algebraically complete (= separably defectless henselian) field of characteristic p > 0. If in addition K^{c}|K is separable, then K is algebraically complete. A slightly different characterization is given by the following consequence of this result, which we also involve in our study of separably tame fields: Let K be a henselian field of characteristic p > 0. If K is a separable–algebraically complete field, then K^{c} is an algebraically complete field, and vice versa.

Furthermore, we prove

Theorem 6 Let K be a separable–algebraically maximal field of characteristic p > 0. If in addition K is inseparably defectless, then K is algebraically complete.

This theorem shows that a valued field is algebraically complete if and only if it is algebraically maximal and inseparably defectless (since the two latter properties are consequences of “algebraically complete”).

The key to these theorems is the closer investigation of immediate extensions of degree p (where 0 < p = the characteristic of the residue field of the given valued field). These extensions are crucial since every finite extension L|K of a henselian field K, if lifted up through a tame extension N|K, will become an extension L.N|N which is a tower of normal extensions of degree p where some of them are immediate if L|K is not defectless. This reduction to normal extensions of degree p is also a main ingredient of the proof of Theorem 4. If the characteristic of K is p, then normal extensions of degree p are either purely inseparable or Artin–Schreier–extensions. We deduce certain normal forms for Artin–Schreier–extensions in order to prove that they are defectless under the given assumptions. After gaining some familiarity with such Artin–Schreier–extensions, the significance of Frobenius closed bases is no longer as surprising as it may appear at first sight.

The characterization of henselian defectless fields which is given by Theorem 6, is very helpful if one wants to construct henselian defectless fields. The “classical” method of just taking maximally valued fields (which are always henselian defectless fields) is too coarse for certain purposes, and we establish an interesting example for this fact. Dealing now with nonperfect fields of characteristic p > 0, we give a valuation theoretical axiom (suggested by L. van den Dries) for valued fields of p–degree 1:

∀x∃y∃x_{0}, ..., x_{p−1} :
(x_{0} = 0 ∨ v(x_{0}) = 0) ∧ x = y^{p} − y + x_{0} + tx_{1}^{p} + ... + t^{p−1}x_{p−1}^{p} .  (3)

We show that it is satisfied by all maximally valued fields of p–degree 1, hence in particular by the power series field IF_{p}((t)). On the other hand, we succeed in constructing an algebraically complete field (K, v) of p–degree 1 which does not satisfy this axiom: there is an element x ∈ K which is not of the form

x = y^{p} − y + x_{0} + tx_{1}^{p} + ... + t^{p−1}x_{p−1}^{p} with x_{0} = 0 ∨ v(x_{0}) = 0  (4)

in K. But we construct an immediate function field of transcendence degree 1 over (K, v) in which x is of the form (4). The model theoretical consequence of this will be discussed later. Here, we should state that it can be deduced that F cannot be a henselian rational function field over K, hence Theorem 2 does not remain true if “tame” (which is equivalent to “algebraically complete and perfect” if char(K) = p > 0) is replaced just by “algebraically complete”.

Finally, we want to mention valuation theoretical results obtained by an application of one of the model theoretical results, namely the AKE–principle for tame fields (which we will describe later).

In the paper [KP], Prestel and the author deduced results on places of function fields over base fields of characteristic 0, using the AKE–principle for henselian fields of residue characteristic 0. By an application of the AKE-principle for tame fields, it is possible to generalize two results of [KP] to the case of perfect base fields (of arbitrary characteristic). The generalization of the Main Theorem of [KP] reads as follows:

Theorem 7 Let F |k be a function field in n variables with perfect base field k. Let Q be a place of F |k and x_{1}, ..., x_{m}, x_{m+1}, ..., x_{m+s} ∈ F. Then there exists a place P of F |k with a finitely generated residue field over k such that

\begin{aligned}x_{i}Q & = & x_{i}P & for & 1 ≤ i ≤ m , \\ v_{Q}(x_{i}) & = & v_{P} (x_{i}) & for & m + 1 ≤ i ≤ m + s . \end{aligned}

Moreover, if r_{1} and d_{1} are natural numbers satisfying

dim(Q) ≤ d_{1} ≤ n − 1 , rr(Q) ≤ r_{1} ≤ n − d_{1} ,

then P can be chosen to satisfy in addition:

(1) dim(P) = d_{1} and F P is a subfield of a purely transcendental extension of the perfect hull of F Q, finitely generated over k,

(2) rr(P) = r_{1} and v_{P} (F) is a finitely generated subgroup of a discrete lexicographic extension of the p–divisible hull of v_{Q}(F), where p = char(k) > 0 or p = 1 if char(k) = 0.

Note that the second important ingredient for the proof of these generalizations is Lemma 3 on relatively algebraically closed subfields of tame fields, which we have already introduced above.

Results on the model theory of valued fields.

The theory of henselian function fields that we have described so far, has interesting applications to the model theory of valued fields. We consider AKE–fields, i.e. valued fields (K, v) that satisfy the following Ax–Kochen–Ershov–principle (“AKE–principle”): if (L, v)|(K, v) is an extension of valued field such that

v(K) ≺_{∃} v(L) and \overline{K} ≺_{∃} \overline{L} ,

i.e. v(K) is existentially closed in v(L) in the language of ordered groups resp. \overline{K} is existentially closed in \overline{L} in the language of fields, then

(K, v) ≺_{∃} (L, v)

i.e. (K, v) is existentially closed in (L, v) in the language of valued fields.

For a submodel N of a model M, the notion “N ≺_{∃} M” means that every existential sentence with constants from N, which holds in M, will also hold in N. An elementary class of valued fields, in which every member is an AKE–field, will be called AKE–class. This notion is basic, in so far as model completeness and completeness results can be derived from it. Indeed, if an elementary class K of valued fields is an AKE–class, then it is model complete, provided that the classes

v(K) = {v(K) | (K, v) ∈ K}
\overline{K} = \{\overline{K} | (K, v) ∈ K\}

of value groups and residue fields are elementary model complete classes. Furthermore, in many cases completeness can be deduced from model completeness by means of prime models.

The following classes of valued fields were shown to be AKE–classes (in literature, this was usually implicitly proved by corresponding model completeness results):

a) The class of all algebraically closed valued fields (by A. Robinson [ROB]). This may be viewed as the beginning of the (explicit) model theory of valued fields. For applications of Robinson’s result in number theory, cf. [ROQ2].

b) The class of all henselian fields of residue characteristic zero (by Ax and Kochen [AK1] and, independently, by Ershov [ER3]). An application of this result to diophantine problems concerning the p–adics was given by Ax and Kochen in [AK1]. A short proof, this time using the notation “existentially closed”, and an application to places of function fields was given by Prestel and the author in [KP].

c) The class of all ℘-adically closed fields (by Ax and Kochen [AK2] for p–adically closed fields; cf. also Prestel and Roquette [PR]).

d) The class of all henselian valued fields of finite ramification (by Ershov [ER5]; cf. also Ziegler [ZIE]).

e) The class of all algebraically maximal Kaplansky–fields (by Ershov [ER4]; cf. also Ziegler [ZIE]).

f) The class of all separable–algebraically maximal Kaplansky–fields of characteristic p > 0; but here one has to add predicates for p–independence to the language of valued fields (by Delon, cf. [DEL]); this means that the AKE–principle is restricted to separable extensions L|K. “separable–algebraically maximal” means that these fields have no separable–algebraic immediate extension. This result is interesting in so far as a weakening of the condition “algebraically maximal” can be compensated by additional predicates.

In addition to these results, we prove:

Theorem 8 Every tame field is an AKE–field.

and: Let (K, v) be a separably tame field and (L, v)|(K, v) a separable extension. If v(K) ≺_{∃} v(L) and \overline{K} ≺_{∃} \overline{L}, then (K, v) ≺_{∃} (L, v).

Direct proofs for all above classes to be AKE–classes can be carried out by means of embedding lemmata. It is deducible from general model theory that (K, v) is existentially closed in (L, v) if and only if (L, v) is embeddable over (K, v) into every |L|^{+}–saturated elementary extension (K, v)^{∗} = (K^{∗}, v^{∗}) of (K, v); by an “embedding of (K, v)” we always mean a valuation–preserving embedding. This makes sense since we assume v(K) ≺_{∃} v(L) and \overline{K} ≺_{∃} \overline{L} which by the same principle as we have sketched for valued fields, shows that v(L) may be viewed as embedded into v^{∗}(K^{∗}) over v(K), and L/v as embedded over K/v into K^{∗}/v^{∗}; here “/v” and “/v^{∗}” denote the respective residue fields. (Note that v^{∗}(L^{∗}) is |L|^{+}–saturated and thus also |v(L)|^{+}–saturated, and an analogue holds for L^{∗}/v^{∗}). Furthermore, it is deducible from general model theory that (L, v) is embeddable in this way if already every finitely generated subextension (L_{0}, v) of (L, v)|(K, v) is embeddable over (K, v) into (K, v)^{∗}. Now a finitely generated extension L_{0}|K is nothing else but a function field over K, so at this point, we encounter anew valued function fields.

On the other hand, it can be shown that a valued field will only satisfy the AKE– principle if it is henselian (more precisely, it must even be algebraically complete. In the above approach this means that (K, v) and also (K, v)^{∗} (being an elementary extension of (K, v) ) will be henselian. In this situation, we ask for an embedding of a henselization (L_{0}, v)^{h} of (L_{0}, v) over (K, v) into (K, v)^{∗}; such embedding would yield the desired embedding of (L_{0}, v) just by restriction. Though we are now dealing with the larger field (L_{0}, v)^{h}, we gain the possibility of using helpful structure theorems for the henselian function field (F, v) = (L_{0}, v)^{h}.

To illustrate this, we want first to sketch the case where the extension (F, v)|(K, v) is immediate: let us assume v(K) = v(F) and \overline{K} = \overline{F}, which trivially yields that the conditions “v(K) ≺_{∃} v(F)” and “\overline{K} ≺_{∃} \overline{F} ” are fulfilled. In addition, let us assume that the transcendence degree of F |K is 1. In order to solve the embedding problem, a suitable property has to be imposed on (K, v) (as we have mentioned already, (K, v) has to be algebraically complete if the AKE–principle should be satisfied in the simple language of valued fields).

Let us now assume that (K, v) is a tame field. By Theorem 2 that we have cited above, we know that (F, v) is a henselian rational function field (K(x), v)^{h} for a suitable x ∈ F which is transcendental over K. Using that (K, v)^{∗} is highly enough saturated by assumption, it is possible to embed (K(x), v) over (K, v) into (K, v)^{∗}. Such embedding can be prolongated to an embedding of K(x)^{h}; this follows from the fact that (K, v)^{∗} is henselian and the universal property of henselizations (they are “minimal” henselian extensions, i.e. they admit a valuation preserving embedding into every other henselian extension field). The henselization may be viewed as a certain closure which is unique up to valuation preserving isomorphism. It is interesting to see that apart from the above approach which uses the henselian rationality, the embedding problem for the mentioned classes a) – e) can as well be solved by using the uniqueness (up to valuation–preserving isomorphism) of suitable closures:

– the algebraic closure in the case of algebraically closed valued fields,

– the henselization for the classes as described in b), c), d),

– the henselization and its maximal algebraic purely wild extensions in the case of Kaplansky–fields. (A similar situation is found for real fields: the uniqueness of the real closure can serve to prove an embedding lemma from which model completeness can be deduced.)

The solution of the special embedding problem yields that tame fields are existentially closed in every immediate extension of transcendence degree 1. On the other hand, we are able to show that this does not remain true if “tame” is replaced by “algebraically complete”. This is done by using the example that we have introduced above. Recall that this example consists of an algebraically complete field (K, v) of p–degree 1, admitting an element x ∈ K which is not of the form (4), together with an immediate function field (F, v) of transcendence degree 1 over K in which x is of the form (4). The fact that x is of the form (4) is an existential sentence with just two constants, namely x and t, both of them being elements of K. Since this sentence holds in F but not in K, we see that (K, v) is not existentially closed in the immediate function field (F, v). Hence we conclude that in general, algebraically complete fields do not satisfy the AKE–principle. Moreover, since every maximally valued field (of p–degree 1) satisfies the axiom (3) as we mentioned above, (K, v) is an example of an algebraically complete field which does not have the property of admitting a maximal immediate extension in which it is existentially closed. On the other hand, the latter property can be shown to imply “algebraically complete”, so it is stronger than “algebraically complete”. It is an open question whether this property allows an algebraic axiomatization.

For the proof of the AKE–principle for tame fields, it is necessary to deal also with the case of henselian function fields without transcendence defect. From Theorem 4 we deduce the AKE–principle for extensions without transcendence defect of algebraically complete fields

Theorem 9 If (L, v) is an extension without transcendence defect of the algebraically complete field (K, v) then the “side conditions”

v(K) ≺_{∃} v(L) and \overline{K} ≺_{∃} \overline{L}

imply

(K, v) ≺_{∃} (L, v) .

Note that this theorem is not only of interest for the proof of the AKE–principle for tame fields: contrary to the case of immediate extensions, it works already for all algebraically complete base fields. There is no elementary axiomatization of the class of extensions without transcendence defect of algebraically complete fields, hence at this point it turns out to be adequate to investigate a single extension rather than all extensions in an elementary class. Moreover, the notion “≺_{∃}” appears to be more adequate than “≺”, since Theorem 9 becomes false if we replace ≺_{∃} by ≺. As an example for this fact, we can use again the algebraically complete field (K, v) which does not satisfy the axiom (3). Indeed, we construct (K, v) to be an extension without transcendence defect of the henselian field (IF_{p}(t), v)^{h} (valued such that v(t) is the smallest positive element in the value group), and this extension even satisfies

v(IF_{p}(t)^{h}) = ZZ ≺ v(K)
\overline{IF_{p}(t)^{h}} = IF_{p} ≺ \overline{K} ;

on the other hand, we show that (IF_{p}(t), v)^{h} satisfies (3). Since (K, v) does not satisfy (3), we have (IF_{p}(t), v)^{h} ⊀ (K, v).

Let us return again to the class of tame fields. This class appears at the first glimpse to be only a slight generalization of the class of all algebraically maximal Kaplansky–fields, since only condition (Kf 3) is dropped. Nevertheless, this turns out to be a qualitative difference in two respects. Firstly, it allows a fruitful application to places of function fields over perfect fields of positive characteristic, whereas to our knowledge, the AKE–principle for algebraically maximal Kaplansky–fields has found no applications to algebra (except for the subclass of algebraically closed valued fields which is already covered by Robinson’s model completeness result). Though the very special form of Kaplansky–fields has evoked a nice structure theory (Kaplansky [KAP]) and a nice Galois theoretical interpretation (cf. [KPR]), it seems that for applications, these valued fields are of minor interest.

Secondly, with tame fields of positive characteristic we encounter the first AKE–class of fields whose maximal immediate extensions are in general not unique up to isomorphism.

We can thus conclude that this property is not necessary for fields to be AKE–fields, and it is adequate to replace its use by a good structure theory for henselian function fields.

Thirdly, our results on tame fields enable us to transfer our approach for tame fields to the classes a) – e) mentioned above. This gives a unified proof for the fact that all these classes are AKE–classes. It is based on criteria for AKE–classes that reflect main properties of tame fields.

Finally, let us point out that model theoretical questions about valued fields have led to the discovery of the interesting new axiom (3) for valued fields, and a whole scheme of axioms related to it, which remain to be studied carefully. Since these axioms show a certain relation to the theory of purely wild extensions of henselian fields, there is some hope that they are fit to replace the insufficient axiom “algebraically complete” in the case of nonperfect fields. But also from the valuation theoretical viewpoint, the investigation of fields satisfying these axioms should be interesting and fruitful.

A valuation theoretical tool of independent interest: approximation types.

We use the new notion of approximation types at several instances. We use them instead of pseudo Cauchy sequences which have been developed by Ostrowski [OS] and Kaplansky [KAP]. The concept of approximation types is more intrinsic than the one of pseudo Cauchy sequences, and they eliminate the arbitrariness that lies in the choice of one single pseudo Cauchy sequence. Moreover, they facilitate computations as well as the comparison of two different elements in a valued field extension concerning their approximation by elements from the base field.

One basic advantage of approximation types in comparison to pseudo Cauchy sequences is that they allow the treatment of elements in valued field extensions with a uniform machinery, whether the extension is immediate or not. For instance, an element in a valued field extension L|K may be the limit of a pseudo Cauchy sequence over K, whereas over an intermediate field K′ of L|K it may not. In this situation, pseudo Cauchy sequences are not adequate for a proper analysis. Furthermore, only pseudo Cauchy sequences which cannot be prolongated, carry the full information. For the formulation of certain properties, one would thus be forced to restrict the scope to those pseudo Cauchy sequences which are maximal, i.e. which cannot be prolongated. In contrast to pseudo Cauchy sequences, the definition of approximation types guarantees in itself the maximality, the full information.

We give a purely valuation theoretical definition of approximation types. They may be viewed as filters which do not forget the values, or as a collection of cosets. We give a full presentation of the theory of approximation type. Since we are trying by this to replace a classical notion, we do not respect the frame that is set by applications in other chapters; in other words, we develop the major part of the theory for its independent interest. It covers the theory of pseudo Cauchy sequences and exceeds it at several points.

Starting from the abstract valuation theoretical definition of approximation types, we show how approximation types can be determined by sets of simple formulas, which may be considered as subsets of 1–types over the valued base field. We also show the relation to the reduct of these types to the language of valued vector spaces over the given base field. This idea is recovered later when a transfer of Kaplansky’s theory of pseudo Cauchy sequences leads to the determination of valued vector spaces whose valuation is fixed by the approximation type. The interpretation of approximation types as subsets of 1– types also enables us to show that every approximation type over a valued base field K is realized by an element x in a valued field extension, so an approximation type is no longer as abstract as appeared by its definition. In certain cases (e.g. if the approximation type is immediate), we may even assume that (K, v) is existentially closed in (K(x), v).

The immediate approximation types are the analogue to pseudo Cauchy sequences. For them we define a degree and the distinction whether they are algebraic or transcendental. Parallel to the theory of Kaplansky, we show that algebraic approximation types are realized in certain immediate algebraic extensions, and that transcendental approximation types are realized in certain immediate transcendental extensions. As to these results, they are merely a transposition of Kaplansky’s work, just a bit more detailed. Then associated minimal polynomials can be associated to algebraic approximation types; these are the potential minimal polynomials for those immediate algebraic extensions in which the approximation type can be realized. In his Lemma 10 in [KAP] which Kaplansky states without using it further, the idea is expressed that for such associated minimal polynomials, interesting normal forms can be found. However, Kaplansky’s hypothesis in Lemma 10 is quite restrictive. Developing Kaplansky’s approach further, we give normal forms in full generality.

One reason for the deployment of approximation types is to distinguish properties of elements in valued field extensions over a common base field by their approximation types over that base field. This principle is used for the classification of immediate extensions of degree p of a henselian base field. A second application is the proof that approximation types of elements in henselizations over a valued field have a special structure. This leads to the following lemma which in turn is used for the study of the completion defect: Let x, y ∈ K̃, y ∉ K. Assume appr(y, K) = appr(x, K) and x ∈ K^{h}. Then

[K^{h}(y) : K^{h}] < [K(y) : K] .

In particular, K(y)|K is not purely inseparable.

A particularly interesting application of the theory of immediate approximation types is the following. In our study of immediate henselian function fields it is a question of central importance whether a given immediate henselian function field of transcendence degree 1 is henselian rational and which are its henselian generators. This amounts to the determination of the degree

[K(x)^{h} : K(y)^{h}]

for elements y ∈ K(x)^{h} \ K which are not necessarily polynomials in x. Under certain hypotheses, we can determine this degree by relating it to the relative approximation degree h_{K}(x : y). Assume that (K, v) is algebraically maximal or just that the approximation type appr(x, K) is transcendental. If y is a polynomial f(x), then by a combination of Kaplansky’s results with Hensel’s Lemma, we get

[K(x)^{h} : K(f(x))^{h}] ≤ h_{K}(x : f(x)) .

If we assume that the rank of (K, v) is 1 and that x does not lie in the completion of (K, v), then for every y ∈ K(x)^{h}, we can prove

[K(x)^{h} : K(y)^{h}] ≤ h_{K}(x : y) ,

and

K(x)^{h} = K(y)^{h} ⟺ h_{K}(x : y) = 1 .

These results and the further theory of relative approximation degrees constitute the main ingredient in the proof of the pull down principle for henselian rationality through tame extensions.

Through all of these examples, it appears that the tool of approximation types is suitable to describe properties and situations which had been remote before. We hope that the approaches and tools as well as the new results that we introduce in this thesis, will help to get access to some of the unsolved problems in valuation theory and the model theory of valued fields.

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